libquadmath/math/expm1q.c

   1 /*                                                      expm1l.c
   2  *
   3  *      Exponential function, minus 1
   4  *      128-bit __float128 precision
   5  *
   6  *
   7  *
   8  * SYNOPSIS:
   9  *
  10  * __float128 x, y, expm1l();
  11  *
  12  * y = expm1l( x );
  13  *
  14  *
  15  *
  16  * DESCRIPTION:
  17  *
  18  * Returns e (2.71828...) raised to the x power, minus one.
  19  *
  20  * Range reduction is accomplished by separating the argument
  21  * into an integer k and fraction f such that
  22  *
  23  *     x    k  f
  24  *    e  = 2  e.
  25  *
  26  * An expansion x + .5 x^2 + x^3 R(x) approximates exp(f) - 1
  27  * in the basic range [-0.5 ln 2, 0.5 ln 2].
  28  *
  29  *
  30  * ACCURACY:
  31  *
  32  *                      Relative error:
  33  * arithmetic   domain     # trials      peak         rms
  34  *    IEEE    -79,+MAXLOG    100,000     1.7e-34     4.5e-35
  35  *
  36  */
  37
  38 /* Copyright 2001 by Stephen L. Moshier
  39
  40     This library is free software; you can redistribute it and/or
  41     modify it under the terms of the GNU Lesser General Public
  42     License as published by the Free Software Foundation; either
  43     version 2.1 of the License, or (at your option) any later version.
  44
  45     This library is distributed in the hope that it will be useful,
  46     but WITHOUT ANY WARRANTY; without even the implied warranty of
  47     MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
  48     Lesser General Public License for more details.
  49
  50     You should have received a copy of the GNU Lesser General Public
  51     License along with this library; if not, write to the Free Software
  52     Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307  USA */
  53
  54
  55
  56 #include <errno.h>
  57 #include "quadmath-imp.h"
  58
  59 /* exp(x) - 1 = x + 0.5 x^2 + x^3 P(x)/Q(x)
  60    -.5 ln 2  <  x  <  .5 ln 2
  61    Theoretical peak relative error = 8.1e-36  */
  62
  63 static const __float128
  64   P0 = 2.943520915569954073888921213330863757240E8Q,
  65   P1 = -5.722847283900608941516165725053359168840E7Q,
  66   P2 = 8.944630806357575461578107295909719817253E6Q,
  67   P3 = -7.212432713558031519943281748462837065308E5Q,
  68   P4 = 4.578962475841642634225390068461943438441E4Q,
  69   P5 = -1.716772506388927649032068540558788106762E3Q,
  70   P6 = 4.401308817383362136048032038528753151144E1Q,
  71   P7 = -4.888737542888633647784737721812546636240E-1Q,
  72   Q0 = 1.766112549341972444333352727998584753865E9Q,
  73   Q1 = -7.848989743695296475743081255027098295771E8Q,
  74   Q2 = 1.615869009634292424463780387327037251069E8Q,
  75   Q3 = -2.019684072836541751428967854947019415698E7Q,
  76   Q4 = 1.682912729190313538934190635536631941751E6Q,
  77   Q5 = -9.615511549171441430850103489315371768998E4Q,
  78   Q6 = 3.697714952261803935521187272204485251835E3Q,
  79   Q7 = -8.802340681794263968892934703309274564037E1Q,
  80   /* Q8 = 1.000000000000000000000000000000000000000E0 */
  81 /* C1 + C2 = ln 2 */
  82
  83   C1 = 6.93145751953125E-1Q,
  84   C2 = 1.428606820309417232121458176568075500134E-6Q,
  85 /* ln 2^-114 */
  86   minarg = -7.9018778583833765273564461846232128760607E1Q;
  87
  88
  89 __float128
  90 expm1q (__float128 x)
  91 {
  92   __float128 px, qx, xx;
  93   int32_t ix, sign;
  94   ieee854_float128 u;
  95   int k;
  96
  97   /* Detect infinity and NaN.  */
  98   u.value = x;
  99   ix = u.words32.w0;
 100   sign = ix & 0x80000000;
 101   ix &= 0x7fffffff;
 102   if (!sign && ix >= 0x40060000)
 103     {
 104       /* If num is positive and exp >= 6 use plain exp.  */
 105       return expq (x);
 106     }
 107   if (ix >= 0x7fff0000)
 108     {
 109       /* Infinity (which must be negative infinity). */
 110       if (((ix & 0xffff) | u.words32.w1 | u.words32.w2 | u.words32.w3) == 0)
 111         return -1.0Q;
 112       /* NaN.  Invalid exception if signaling.  */
 113       return x + x;
 114     }
 115
 116   /* expm1(+- 0) = +- 0.  */
 117   if ((ix == 0) && (u.words32.w1 | u.words32.w2 | u.words32.w3) == 0)
 118     return x;
 119
 120   /* Minimum value.  */
 121   if (x < minarg)
 122     return (4.0/HUGE_VALQ - 1.0Q);
 123
 124   /* Avoid internal underflow when result does not underflow, while
 125      ensuring underflow (without returning a zero of the wrong sign)
 126      when the result does underflow.  */
 127   if (fabsq (x) < 0x1p-113Q)
 128     {
 129       math_check_force_underflow (x);
 130       return x;
 131     }
 132
 133   /* Express x = ln 2 (k + remainder), remainder not exceeding 1/2. */
 134   xx = C1 + C2;                 /* ln 2. */
 135   px = floorq (0.5 + x / xx);
 136   k = px;
 137   /* remainder times ln 2 */
 138   x -= px * C1;
 139   x -= px * C2;
 140
 141   /* Approximate exp(remainder ln 2).  */
 142   px = (((((((P7 * x
 143               + P6) * x
 144              + P5) * x + P4) * x + P3) * x + P2) * x + P1) * x + P0) * x;
 145
 146   qx = (((((((x
 147               + Q7) * x
 148              + Q6) * x + Q5) * x + Q4) * x + Q3) * x + Q2) * x + Q1) * x + Q0;
 149
 150   xx = x * x;
 151   qx = x + (0.5 * xx + xx * px / qx);
 152
 153   /* exp(x) = exp(k ln 2) exp(remainder ln 2) = 2^k exp(remainder ln 2).
 154
 155   We have qx = exp(remainder ln 2) - 1, so
 156   exp(x) - 1 = 2^k (qx + 1) - 1
 157              = 2^k qx + 2^k - 1.  */
 158
 159   px = ldexpq (1.0Q, k);
 160   x = px * qx + (px - 1.0);
 161   return x;
 162 }